Average Error: 31.6 → 0.3
Time: 3.5s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03391785124353622477011427349680161569268:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\ \mathbf{elif}\;x \le 0.03384107969428518797316840505118307191879:\\ \;\;\;\;\frac{1}{720} \cdot {x}^{4} + \left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{\cos x - 1}{x}}{x}\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03391785124353622477011427349680161569268:\\
\;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\

\mathbf{elif}\;x \le 0.03384107969428518797316840505118307191879:\\
\;\;\;\;\frac{1}{720} \cdot {x}^{4} + \left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{-1 \cdot \frac{\cos x - 1}{x}}{x}\\

\end{array}
double f(double x) {
        double r20234 = 1.0;
        double r20235 = x;
        double r20236 = cos(r20235);
        double r20237 = r20234 - r20236;
        double r20238 = r20235 * r20235;
        double r20239 = r20237 / r20238;
        return r20239;
}
↓
double f(double x) {
        double r20240 = x;
        double r20241 = -0.033917851243536225;
        bool r20242 = r20240 <= r20241;
        double r20243 = 1.0;
        double r20244 = cos(r20240);
        double r20245 = r20243 - r20244;
        double r20246 = 1.0;
        double r20247 = r20246 / r20240;
        double r20248 = r20247 / r20240;
        double r20249 = r20245 * r20248;
        double r20250 = 0.03384107969428519;
        bool r20251 = r20240 <= r20250;
        double r20252 = 0.001388888888888889;
        double r20253 = 4.0;
        double r20254 = pow(r20240, r20253);
        double r20255 = r20252 * r20254;
        double r20256 = 0.5;
        double r20257 = 0.041666666666666664;
        double r20258 = 2.0;
        double r20259 = pow(r20240, r20258);
        double r20260 = r20257 * r20259;
        double r20261 = r20256 - r20260;
        double r20262 = r20255 + r20261;
        double r20263 = -1.0;
        double r20264 = r20244 - r20243;
        double r20265 = r20264 / r20240;
        double r20266 = r20263 * r20265;
        double r20267 = r20266 / r20240;
        double r20268 = r20251 ? r20262 : r20267;
        double r20269 = r20242 ? r20249 : r20268;
        return r20269;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.033917851243536225

    1. Initial program 1.3

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied associate-/r*0.5

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity0.5

      \[\leadsto \frac{\frac{1 - \cos x}{x}}{\color{blue}{1 \cdot x}}\]

    6. Applied div-inv0.5

      \[\leadsto \frac{\color{blue}{\left(1 - \cos x\right) \cdot \frac{1}{x}}}{1 \cdot x}\]

    7. Applied times-frac0.6

      \[\leadsto \color{blue}{\frac{1 - \cos x}{1} \cdot \frac{\frac{1}{x}}{x}}\]

    8. Simplified0.6

      \[\leadsto \color{blue}{\left(1 - \cos x\right)} \cdot \frac{\frac{1}{x}}{x}\]

    if -0.033917851243536225 < x < 0.03384107969428519

    1. Initial program 62.3

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]
    3. Using strategy rm

    4. Applied associate--l+0.0

      \[\leadsto \color{blue}{\frac{1}{720} \cdot {x}^{4} + \left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)}\]

    if 0.03384107969428519 < x

    1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied associate-/r*0.4

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}}\]

    4. Taylor expanded around -inf 0.4

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{\cos x - 1}{x}}}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03391785124353622477011427349680161569268:\\ \;\;\;\;\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{x}\\ \mathbf{elif}\;x \le 0.03384107969428518797316840505118307191879:\\ \;\;\;\;\frac{1}{720} \cdot {x}^{4} + \left(\frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{\cos x - 1}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))